Methods and systems for conservative extraction of over-represented extensible motifs

ABSTRACT

Methods and systems of extracting extensible motifs from a sequence include assigning a significance to extensible motifs within the sequence based upon a syntactic and statistical analysis, and identifying extensible motifs having a significance that exceeds a predetermined threshold.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention generally relates to a method and system forextraction of extensible motifs. More particularly, the presentinvention relates to a method and system for extraction of extensiblemotifs using combinatorial and statistical pruning.

2. Description of the Related Art

The discovery of a motif in a biosequence is frequently torn between therigidity of the model on the one hand and the abundance of candidates onthe other. In particular, the variety of motifs described by stringsthat include “don't care” patterns escalates exponentially with thelength of the motif, and this only gets worse if a “don't care” isallowed to stretch up to some prescribed maximum length. Thiscircumstance tends to generate daunting computational burdens, and oftengives rise to tables that are impossible to visualize and digest. Thisis unfortunate, as it seems to preclude precisely those massive analysesthat have become conceivable with the increasing availability of massiveamount of genomic and protein data. While part of the problem isendemic, another part of it seems rooted in the variouscharacterizations offered for the notion of a motif, that are typicallybased either on syntax or on statistics alone.

The discovery of motifs in bio-sequences is attracting increasinginterest due to the perceived multiple implication of motifs inbiological structure and function. The approaches to motif discovery maybe partitioned into two main classes. In the first class, the samplestring is tested for occurrences of motifs in a family of a prioridefined, abstract models or templates. The second class of approachesassumes that the search may be limited to substrings in the sample or tosome more or less controlled neighborhood of those substrings. Theapproaches in the first class are more rigorously justifiable, but oftenpose daunting computational burdens. Those in the second class tend tobe computationally viable but rest on more shaky methodological grounds.

The characterizations offered for the notion of a motif could bepartitioned roughly into statistical and syntactic. In a typicalstatistical characterization, a motif is a sequence of m positions suchthat at each position each character from (some subset of) the alphabetmay occur with a given probability or weight. This is often described bya suitable matrix or profile, where columns correspond to positions androws to alphabet characters. The lineage of syntactic characterizationscould be ascribed to the theory of error correcting codes: a motif is apattern w of length m and an occurrence of it is any string at adistance of d, the distance being measured in terms of errors of acertain type. For example, we can have only substitutions in a Hammingvariant, substitutions and indels in a Levensthein variant, and so on.Syntactic characterizations enable us to describe the model of a motif,or a realization of it, or both, as a string or simple regularexpression over an extension of the input alphabet Σ, e.g., over Σ∪{.},where “.” denotes the “don't care” character.

Irrespective of the particular model or representation chosen, the tenetof motif discovery equates over-representation of a motif with surpriseand hence with interest. Thus, any motif discovery algorithm mustultimately weigh motifs against some threshold, based on a score thatcompares empirical and expected frequency, perhaps with somenormalization. The departure of a pattern w from expectation is commonlymeasured by so-called z-scores, which have the form:

$\begin{matrix}{{z(w)} = \frac{{f(w)} - {E(w)}}{N(w)}} & (1)\end{matrix}$

where:

f(w)>0 represents a frequency;

E(w)>0 represents an expectation; and

N(w)>0 is the expected value of some function of w.

For given z-score function, set of patterns W, and real positivethreshold T, patterns such that z(w)>T or z(w)<−T are respectivelydubbed over- or under-represented, or simply surprising. The problem isthat the number of patterns extracted in this way may escalate quiterapidly, a circumstance that seems to preclude precisely those massiveanalyses that have become conceivable with the increasing availabilityof whole genomes.

Large-scale statistical tables may not only impose an unbearablecomputational burden. They are also impractical to visualize and use, acircumstance that may defy the purpose of building them in the firstplace.

A little reflection establishes how an exponential build-up may takeplace. Assume that on the binary alphabet both aabaab and abbabb areasserted as reflections of candidate interesting motifs. A concisedescription of this motif is a.ba.b, with “.” denoting the don't care,and then look for further occurrences of this motif. By this, however,the spurious patterns aababb and abbaab are also annexed.

A similar problem presents itself in the approaches that resort to theprofiles or the weighted matrices previously mentioned. Even settingaside computational aspects, tables that are too large at the outset runthe risk of saturating the visual bandwidth of a user. In this spirit,approaches that limit the number of patterns to be considered from thestart may provide a more significant throughput, even in comparison withexhaustive methods.

SUMMARY OF THE INVENTION

In view of the foregoing and other exemplary problems, drawbacks, anddisadvantages of the conventional methods and structures, an exemplaryfeature of the present invention is to provide methods and structures inwhich the significance of extensible motifs are identified by acombination of syntactic and statistical analysis.

In a first exemplary aspect of the present invention, a method ofextracting extensible motifs from a sequence includes assigning asignificance to extensible motifs within the sequence based upon asyntactic and statistical analysis, and identifying extensible motifshaving a significance that exceeds a predetermined threshold.

In a second exemplary aspect of the present invention, a system forextracting extensible motifs from a sequence includes means forassigning a significance to extensible motifs within the sequence basedupon a syntactic and statistical analysis, and means for identifyingextensible motifs having a significance that exceeds a predeterminedthreshold.

In a third exemplary aspect of the present invention a program isembodied in a computer readable medium executable by a digitalprocessing unit. The program includes instructions for assigning asignificance to extensible motifs within the sequence based upon asyntactic and statistical analysis, and instructions for identifyingextensible motifs having a significance that exceeds a predeterminedthreshold.

The inventors regard the motif discovery process as distributed into twostages, where the first stage unearths motifs endowed with a certain setof properties and the second filters out the interesting ones. Since theredundancy builds up in the first stage, it is there that the inventorsdecided to look for possible ways of reducing the unnecessarythroughput. Since over-representation is measured by a score, it isdesirable to find ways to neglect candidate motifs that cannot possiblymake it to the top list, and ideally spot such motifs before they areeven computed. Counterintuitive as it might look, the inventorsdiscovered that such a possibility may be offered by certain attributesof “saturation” that combine in a unique way the syntactic structure andthe list of occurrences or frequency for a motif.

With solid words, for example, it is known that in the worst case thenumber of distinct substrings in a string can be quadratic in the lengthof that string. Yet, if the substrings are partitioned into buckets byputting in the same bucket strings that have exactly the same set ofoccurrences, then only the number of buckets which are linear in thetextstring are needed.

Similar linear bounds may be established for special classes of rigidmotifs containing “don't cares”. When combined with intervals of scoremonotonicity, properties of this kind support the global detection ofunusual words of any length in overall linear space. Some of theseconservative scoring techniques were extended recently to rigid motifswith a prescribed maximum number of mismatches or don't care.

An exemplary method and system in accordance with the present inventioncombines a structure of a motif pattern, as described by its syntacticspecification, with a statistical measure of its occurrence count.

An exemplary embodiment of the present invention characterizes a patternrigidly, and conjugates structure and set of occurrences. This resultsin a definition of motif that lends itself to a natural notion ofmaximality, thereby embodying statistics and structure in one measure ofsurprise. This is unlike all previous approaches that consider structureand statistics as separate features of a pattern.

An exemplary embodiment of the present invention provides a powerfulsyntactic mechanism for eliminating unimportant motifs before theirscore is computed. As explained above, for the class of over-representedmotifs, the non-maximal motifs are not more surprising than the maximalmotifs.

In an exemplary embodiment of the present invention, a combination ofappropriate saturation conditions (expressed in terms of minimum numberof don't cares compatible with a given list of of occurrences) and themonotonicity of probabilistic scores over regions of constant frequencyprovide significant parsimony in the generation and testing of candidateover-represented motifs.

The advantages of exemplary embodiments of the present invention aredocumented by experimental results obtained when specifically targetingprotein sequence families. In all cases tested, the motif reported in adatabase of protein families and domains known as “PROSITE” (a databaseof protein families and domains that includes biologically significantsites, patterns and profiles that help to reliably identify to whichknown protein family (if any) a new sequence belongs) as most importantin terms of functional/structural relevance emerges among the top thirtyextensible motifs returned by an exemplary embodiment of the presentinvention, often right at the top.

Of equal importance seems the fact that the sets of all surprisingmotifs returned in each experiment are extracted faster and come in muchmore manageable sizes using an exemplary embodiment of the presentinvention than would be obtained in the absence of saturationconstraints.

An exemplary embodiment of the present invention provides acharacterization of extensible motifs in the definition of whichstructural or syntactic properties and occurrence statistics are solidlyintertwined.

An exemplary embodiment of the present invention provides a combinationof saturation conditions (expressed in terms of minimum number of don'tcares compatible with a given list of occurrences) and monotonicity ofscores which provides significant parsimony in the generation andtesting of candidate over-represented motifs.

An exemplary embodiment of the present invention isolates as candidatesurprising motifs only the members of an previously well identified setof “maximally saturated” patterns. By this set being identifiable apriori, the embodiment includes motifs in the set that are known beforeany score is computed. By neglecting the motifs other than those in theset of “maximally saturated” patterns, surprising motifs are notoverlooked. In fact, any such motif: (i) is embedded in one of thesaturated motifs, and (ii) does not achieve a larger score than thelatter (hence, computing its score and publishing it explicitly wouldtake more time and space but not add information).

An exemplary embodiment of the present invention applies to extensiblepatterns a philosophy that was previously applied only to rigid motifsby solid words and by words of some specified fixed length affected by aspecified maximum number of errors. The invention enables a transitionfrom rigid to extensible motifs, thereby providing methods and systemsthat extract and weigh extensible motifs.

The inventors illustrate below the merits of exemplary embodiments ofthe present invention on families of protein sequences. In all casestested, the motif reported in PROSITE as most important in terms offunctional/structural relevance emerges either at the top or among thetop ten or so of the output list that is provided by an exemplaryembodiment of the present invention.

These and many other advantages may be achieved with the presentinvention.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other exemplary purposes, aspects and advantages willbe better understood from the following detailed description of anexemplary embodiment of the invention with reference to the drawings, inwhich:

FIG. 1 illustrates an exemplary hardware/information handling system 100for incorporating the present invention therein;

FIG. 2 illustrates a signal bearing medium 200 (e.g., storage medium)for storing steps of a program of a method according to the presentinvention; and

FIG. 3 illustrates a flowchart of an exemplary method in accordance withthe present invention.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS OF THE INVENTION

Referring now to the drawings, and more particularly to FIGS. 1-3, thereare shown exemplary embodiments of the method and structures of thepresent invention.

FIG. 1 illustrates a typical hardware configuration of an informationhandling/computer system for use with the invention and which preferablyhas at least one processor or central processing unit (CPU) 111.

The CPUs 111 are interconnected via a system bus 112 to a random accessmemory (RAM) 114, read-only memory (ROM) 116, input/output (I/O) adapter118 (for connecting peripheral devices such as disk units 121 and tapedrives 140 to the bus 112), user interface adapter 122 (for connecting akeyboard 124, mouse 126, speaker 128, microphone 132, and/or other userinterface device to the bus 112), a communication adapter 134 forconnecting an information handling system to a data processing network,the Internet, an Intranet, a personal area network (PAN), etc., and adisplay adapter 136 for connecting the bus 112 to a display device 138and/or printer.

In addition to the hardware/software environment described above, adifferent aspect of the invention includes a computer-implemented methodfor performing the methods described herein. As an example, an exemplarymethod in accordance with the present invention may be implemented inthe particular environment discussed above.

Such a method may be implemented, for example, by operating a computer,as embodied by a digital data processing apparatus, to execute asequence of machine-readable instructions. These instructions may residein various types of signal-bearing media.

This signal-bearing media may include, for example, a RAM containedwithin the CPU 111, as represented by the fast-access storage forexample. Alternatively, the instructions may be contained in anothersignal-bearing media, such as a magnetic data storage diskette 200 (FIG.2), directly or indirectly accessible by the CPU 111.

Whether contained in the diskette 200, the computer/CPU 111, orelsewhere, the instructions may be stored on a variety ofmachine-readable data storage media, such as DASD storage (e.g., aconventional “hard drive” or a RAID array), magnetic tape, electronicread-only memory (e.g., ROM, EPROM, or EEPROM), an optical storagedevice (e.g. CD-ROM, WORM, DVD, digital optical tape, etc.), paper“punch” cards, or other suitable signal-bearing media includingtransmission media such as digital and analog and communication linksand wireless. In an illustrative embodiment of the invention, themachine-readable instructions may comprise software object code,compiled from a language such as “C”, etc.

To proceed with a formal definition of the concepts highlighted above,let s be a sequence of sets of characters from an alphabet Σ∪{.}, where‘.’ ∉Σ denotes a don't-care (dot, for short) and the rest are solidcharacters. The inventors use σ to denote a singleton character or asubset of Σ. For character (sets) e₁ and e₂, the inventors write e₁

e₂ if and only if e₁ is a dot or e₁ ⊂e₂. Allowing for spacers in astring is what makes it extensible. Such spacers are indicated byannotating the dot characters. Specifically, an annotated “.” characteris written as .^(α) where α is a set of positive integers {α₁, α₂, . . ., α_(k)} or an interval α=[α_(l), α_(u)], representing all integersbetween α_(l) and α_(u) including α₁ and α_(u). Whenever defined, d willdenote the maximum number of consecutive dots allowed in a string. Insuch cases, for clarity of notation, the inventors use the extensiblewild card denoted by the dash symbol “-” instead of the annotated dotcharacter, .^([1,d]) in the string. Note that ‘-’∉Σ. Thus a string ofthe form a.^([1,d]) b will be simply written as a-b.

A motif m is extensible if it contains at least one annotated dot,otherwise m is rigid. Given an extensible string m, a rigid string m′ isa realization of m if each annotated dot .^(α) is replaced by l εα adots. The collection of all such rigid realizations of m is denoted byR(m). A rigid string m occurs at position l on s if m[j]

s[l+j−1] holds for 1≦j≦|m|. An extensible string m occurs at position lin s if there exists a realization m′ of m that occurs at l. Note thanan extensible string m could possibly occur a multiple number of timesat a location on a sequence s.

For a sequence s and positive integer k, k≦|s|, a string (extensible orrigid) m is a motif of s with |m|>1 and location list L_(m)=(l₁, l₂ , .. . , l_(p)), if both m[1] and m[|m|] are solid and L_(m) is the list ofat all and only the occurrences of m in s. Given a motif m let m[j₁],m[j₂], . . . m[j_(l)] be the l solid elements in the motif m. Then thesub-motifs of m are given as follows: for every j_(i), j_(t), thesub-motif m[j_(i), . . . , j_(t)] is obtained by dropping all theelements before (to the left of) j_(i) and all elements after (to theright of) j_(t) in m. The inventors also note that m is a condensationfor any of its sub-motifs. The inventors are interested in motifs forwhich any condensation would disrupt the list of occurrences. Formally,let m₁, m₂, . . . , m_(k) be the motifs in a string s. A motif m_(i) ismaximal in length if there exists no m_(l), l≠i with |L_(m) _(i)|=|L_(m) _(t) | and m_(i) is a sub-motif of m_(l). A motif m_(i) ismaximal in composition if no dot character of m_(i) can be replaced by asolid character that appears in all the locations in L_(m). A motifm_(i) is maximal in extension if no annotated dot character of m_(i) canbe replaced by a fixed length substring (without annotated dotcharacters) that appears in all the locations in L_(m). A maximal motifis maximal in composition, in extension and in length.

Expectations and Scores

Beginning by deriving some simple expressions for the probability p_(m)of an extensible motif m under stationary, iid assumptions. Let m be anextensible motif generated by a stationary, iid source which emits σεΣwith probability p_(σ). Consider the set R(m) of all possiblerealizations of m. Each realization is a string over Σ∪ {.}. For aspecific realization m, its probability p _(m) is given by:

$\begin{matrix}{{p_{\overset{\_}{m}} = {\prod\limits_{\sigma \in \Sigma}\left( p_{\sigma} \right)^{j_{\sigma}}}},} & (2)\end{matrix}$

where:

j_(σ) is the number of times σ appears in m.

Thus, the dot has implicitly probability 1.

An extensible motif is degenerate if it can possibly have multipleoccurrences at a site i on the input s.

Lemma 1 Let m be an extensible non-degenerate motif generated by astationary, iid source which emits (σεΣ) with probability p_(σ). Letj_(σ) be the number of times σ appears in m and let e be the number ofannotated dots in m with annotations α₁, α₂, . . . α_(e). Then

$\begin{matrix}{p_{m} = {\prod\limits_{\sigma \in \Sigma}{\left( p_{\sigma} \right)^{j_{\sigma}}{\prod\limits_{i = 1}^{e}{\alpha_{i}}}}}} & (3)\end{matrix}$

Proof. Since the motif is non-degenerate, by the definition ofrealization of a motif,

$\begin{matrix}{p_{m} = {\sum\limits_{\overset{\_}{m} \in {R{(m)}}}\left( p_{\overset{\_}{m}} \right)}} & (4)\end{matrix}$

Hence we need to compute p _(m) where m is a rigid motif. Assume m is arigid motif with no dot characters. By the i.i.d. assumption:

$\begin{matrix}{p_{\overset{\_}{m}} = {\prod\limits_{\sigma \in \Sigma}\left( p_{\sigma} \right)^{j_{\sigma}}}} & (5)\end{matrix}$

Next, consider m to be a rigid motif with possibly some dot characters.Again, clearly:

$\begin{matrix}{p_{\overset{\_}{m}} = {\prod\limits_{\sigma \in \Sigma}\left( p_{\sigma} \right)^{j_{\sigma}}}} & (6)\end{matrix}$

In other words, only the solid characters contribute non-trivially tothe computation of p _(m) . Hence, if m is not rigid:

$\begin{matrix}{p_{m} = {{{R(m)}}{\prod\limits_{\sigma \in \Sigma}\left( p_{\sigma} \right)^{j_{\sigma}}}}} & (7) \\{But} & \; \\{{{R(m)}} = {\prod\limits_{i = 1}^{e}{\alpha_{i}}}} & (8)\end{matrix}$

hence the result.

Corollary 2 If m is a non-degenerate extensible motif where each m[i] isa set of (homologous) characters, then

$\begin{matrix}{p_{m} = {\prod\limits_{{{m{\lbrack i\rbrack}} \neq {‘.’}},{‘\text{-}’}}{\left( {\sum\limits_{\sigma \in {m{\lbrack i\rbrack}}}p_{\sigma}} \right){\prod\limits_{i = 1}^{e}{\alpha_{i}}}}}} & (9)\end{matrix}$

Let M^(s) denote a set of strings that has only the solid characters ofat least s occurrences of m. For example, consider the motif a-b withrealizations a.b, a..b and a...b. Then:M ¹={a.b,a..b,a...b}  (10)

since m occurs once on each mεM¹M²={a.bb,a..bb,a.b.b}  (11)

since m occurs twice on each mεM²:M³={a.bbb}  (12)

since m occurs three times on m εM³.

Corollary 3 Let m be a degenerate (possibly with multiple occurrences ata site) extensible motif, and let:p _(m) _(k) =Σ_(m′εM) _(k+1) p_(m′)  (13)

then

$\begin{matrix}{p_{m} = {\sum\limits_{k = 0}^{r - 1}{\left( {- 1} \right)^{k}\left( p_{m^{k + 1}} \right)}}} & (14)\end{matrix}$

This follows directly from the inclusion-exclusion principle.

Notice that for a degenerate motif, Equation (2) is the zero-th orderapproximation of Equation (13). The first order approximation is:p_(m)≈p_(m) ₁ −p_(m) ₂   (15)

and the second order approximation isp_(m)≈p_(m) ₁ −p_(m) ₂ +p_(m) ₃   (16)

and so on. Using Bonferroni's inequalities, a kth order approximation ofp_(m) is an over-estimate of p_(m), if k is odd.

Next, the form of p_(m) for a non-degenerate motif when input m isassumed to be generated by a Markov chain is obtained. For thederivation below, we assume the Markov chain has order 1. For furtherdiscussion, we introduce the following definition.

Definition 4 (cell<σ₁,σ₂,l >, C(m)) A substring {circumflex over (m)},on m is a cell, that begins and ends in solid characters with onlynon-solid intervening characters: σ₁, at the start and σ₂ at the endposition and l is the number of intervening un-annotated dot characters.If the intervening character is the extensible character, then l takes avalue of −1. For convenience, the cell is represented by the triplet<σ₁, σ₂, l >. C(m) is the collection of all such cells of m.

For example,C(ab..c.d−g)={<a,b,0>,<b,c,2>,<c,d,1>,<d,g,−1>}  (17)

Let p_(σ) ₁ _(,σ) ₂ ^((k)) denote the probability of moving from σ₁ toσ₂ in k steps. Let s be a stationary, irreducible, aperiodic Markovchain of order 1 with state space Σ (|Σ|<∞). Further, π_(σ) is theequilibrium probability of σεΣ and the (|Σ|×|Σ|) transition probabilitymatrix P[i,j] is defined as p_(σ) _(i) _(,σ) _(j) ⁽¹⁾. For a rigid motifm, for each cell <σ₁,σ₂, l >ε C( m) is such that l ≧0. It is easy to seethat when l ≧0, the cell represents the (l +1)-step transitionprobability given by P^(l +1), i.e.:p _(σ) ₁ _((.)l σ) ₂ =P ^(l) [σ₁,σ₂].   (18)

Thus for a rigid motif m,

$\begin{matrix}{p_{\overset{\_}{m}} = {\pi_{\overset{\_}{m}{\lbrack 1\rbrack}}{\prod\limits_{{\langle{\sigma_{1},\sigma_{2},\ell}\rangle} \in {C{(\overset{\_}{m})}}}{{P^{\ell}\left\lbrack {\sigma_{1},\sigma_{2}} \right\rbrack}.}}}} & (19)\end{matrix}$

From now on, let u and v be two motifs such that v is a condensation ofu, and consider an arbitrary sequence of consecutive unitexpansions—consisting each of inserting a character or character set atsome position, or replacing a dot character with a solid character orcharacter set—that transforms u into v. A score z is monotonic for u andv if the value of z is always either increasing or decreasing over anysuch expansion. The key observation here is that, under mostprobabilistic settings, the probability of a condensation v of u obeysp_(v)≦p_(u). This is almost immediate under iid distribution, as thefollowing claim shows.

Theorem 5 Let v and u be possibly degenerate extensible motifs under theiid model and let v be a condensation of u. Then, there is an integer{circumflex over (p)}≦1 such that:p_(v)=p_(u){circumflex over (p)}.   (20)

Proof. It is enough to consider the case of a unit condensation, i.e.,where v has one more solid character than u. The claim holds triviallywhen the extra character is introduced as a prefix, an infix, or asuffix of u. In fact, in any such case the probability of the extracharacter multiplies each term of Equation (6), whence the wholeprobability as well.

Consider next the case where the solid character in v substitutes adon't care of u. We begin by describing an alternate way to computep_(u). With l denoting the length of a longest string in R(u), computethe set of all strings over Σ^(l) and store them consecutively row-wisein a table. Compute, for each row, the probability of the string in thatrow, which is the product of the probabilities of the individualcharacters (the sum of all row probabilities is 1). Consider now therealizations in R(u) in succession. Check each realization against everyrow of the table; wherever the two match, mark the row if it had notbeen already marked. Let R be the set of rows that are marked at theoutset. Clearly, adding up the probabilities of the rows in R yieldsp_(u). Consider now the set of rows that would be similarly involved inthe computation of p_(v). This must be a subset of R, whencep_(v)≦p_(u).

With Markov processes, the intuition at the basis is that if we splitthe transition probability into two consecutive segments then we have:P ^(l) [σ₁,σ₂]=Σ_(σ) _(k) _(εΣ) P ^(l) ¹ [σ₁,σ_(k) ]×P ^(l) ²[σ_(k),σ₂]  (21)where:l =l _(1 +l) ₂.   (22)Since all:P^(l) [σ_(i),σ_(j)]≧0   (23)

then any specific character (or alphabet subset) acting as a bottleneckyields:P ^(l) [σ₁,σ₂ ]≦P ^(l) ¹ [σ₁,σ_(k) ]×P ^(l) ² [σ_(k),σ₂].   (24)

Theorem 6 If:f(u)=f(v)>0   (25)N(v)<N(u),   (26)andE(v)/N(wv)≦E(u)/N(u),   (27)

then

$\begin{matrix}{\frac{{f(v)} - {E(v)}}{N(v)} > \frac{{f(u)} - {E(u)}}{N(u)}} & (28)\end{matrix}$

Proof. Multiplying both terms by N(v)/E(v) and using the assumption:f(v)=f(u)>0   (29)

we get, after rearrangement:

$\begin{matrix}{{\frac{f(u)}{E(v)}\left( {1 - \frac{N(v)}{N(u)}} \right)} > {1 - \frac{{E(u)}{N(v)}}{{E(v)}{N(u)}}}} & (30)\end{matrix}$

Since:0<N(v)/N(u)<1   (31)

then the left hand side is always positive. The right hand size isalways negative or zero.

When N(u) is the square root of the variance, the z-score takes up theform:

$\begin{matrix}{{z(u)} = \frac{{f(u)} - {E(u)}}{\sqrt{{Var}(u)}}} & (32)\end{matrix}$

In the Bernoulli model, for instance, this variance results in √{squareroot over (np_(u)(1−p_(u)))}. Let p_(m) be the probability of the motifm occurring at any location i on the input string s with n=|s| and letk_(m) be the observed number of times it occurs on s. When it can beassumed that the occurrence of a motif m at a site is an iid process,for large n and k_(m)<<n we have:

$\begin{matrix}{\frac{k_{m} - {n\; p_{m}}}{\sqrt{n\;{p_{m}\left( {1 - p_{m}} \right)}}}->{N\left( {0,1} \right)}} & (33)\end{matrix}$

Theorem 7 Let u and v be motifs generated with respective probabilitiesp_(u) and:p_(v)=p_(u){circumflex over (p)}  (34)

according to an iid process. If f(u)=f(v) and p_(u)<½ then:

$\begin{matrix}{\frac{{f(v)} - {E(v)}}{\sqrt{{E(v)}\left( {1 - p_{v}} \right)}} > \frac{{f(u)} - {E(u)}}{\sqrt{{E(u)}\left( {1 - p_{u}} \right)}}} & (35)\end{matrix}$

Proof. The functions:N(u)=√{square root over (E(u)(1−p _(u)))}{square root over (E(u)(1−p_(u)))}  (36)

and E(u)/N(u) satisfy the conditions of Theorem 6. First, E(v)<E(u).Indeed, since:|v|−|u|/(n−|u|+1)>0,   (37)

$\begin{matrix}{\frac{E(v)}{E(u)} = {\frac{\left( {n - {v} + 1} \right)p_{v}}{\left( {n - {u} + 1} \right)p_{u}} = {{\left( {1 - \frac{{v} - {u}}{n - {u} + 1}} \right)\hat{p}} < \hat{p} < 1.}}} & (38)\end{matrix}$

Next, we study the ratio:

$\begin{matrix}{\left( \frac{N(v)}{N(u)} \right)^{2} = {{\left( {1 - \frac{{v} - {u}}{n - {u} + 1}} \right)\frac{p_{v}\left( {1 - p_{v}} \right)}{p_{u}\left( {1 - p_{u}} \right)}} < \frac{p_{v}\left( {1 - p_{v}} \right)}{p_{u}\left( {1 - p_{u}} \right)}}} & (39)\end{matrix}$

The concave product p_(u)(1−p_(u)) reaches its maximum for p_(u=)½.Since we assume p_(u)<½, the rightmost term is smaller than one. Themonotonicity of N(u) is satisfied.

Finally, we prove that also E(u)/N(u) is monotonic, i.e., that:E(v)/N(v)≦E(u)/N(u),   (40)

which is equivalent to:

$\begin{matrix}{{\frac{E(v)}{E(u)}\frac{1 - p_{u}}{1 - p_{v}}} \leq 1} & (41)\end{matrix}$

but E(v)/E(u)<1 by hypothesis and (1−p_(u))/(1−p_(v))<1 sincep_(u)>p_(v).

In conclusion, an exemplary embodiment of the present invention mayrestrict the z-score computation to classes of maximal motifs, i.e.,only compute the z-score for the maximally saturated motif among thosein each class of motifs sharing the same list of occurrences.

An exemplary embodiment of the present invention pairwise iteratescombinations of segments of maximal extensible motifs, and prunes thosepairings that are found to not be viable. The input may be a string s ofsize n and two positive integers, K and D. The extensibility parameter Dis interpreted in the sense that up to D (or 1 to D) number of dotcharacters between two consecutive solid characters are allowed. Theoutput is all maximal extensible (with D spacers) patterns that occur atleast K times in s.

Incidentally, an exemplary embodiment of the present invention mayextract rigid motifs as a special case. For this, it suffices tointerpret D as the maximum number of dot characters between twoconsecutive solid characters.

An exemplary embodiment converts the input into a sequence of possiblyoverlapping cells (see Definition 4). A maximal extensible pattern is asequence of cells.

Initialization Phase

The cell is the smallest extensible component of a maximal pattern andthe string can be viewed as a sequence of overlapping cells. If no don'tcare characters are allowed in the motifs then the cells arenon-overlapping. An initialization phase in accordance with an exemplaryembodiment of the present invention may:

1) Construct patterns that have exactly two solid characters in them andseparated by no more than D spaces or “.” characters. This may be doneby scanning the strings from left to right.

Further, for each location this exemplary embodiment may store start andend positions of the pattern. For example, if s=abzdabyxd and K=2, D=2,then all the patterns generated at this step are: ab, a.z, a..d, bz,b.d, b..a, zd, z.a, z..b, da, d.b, d..y, a.y, a..x, by, b.x, b..d, yx,y.d, xd, each with its occurrence list. Thus L_(ab)={(1,2),(5,6)},L_(a.z)={(1,3)} and so on.

2) The extensible cells may be constructed by combining all the cellswith a dot character and the same start and end solid characters. Thelocation list is updated to reflect the start and end position of eachoccurrence. Continuing the previous example, b-d is generated at thisstep with L_(b-d={()2,4),(6,9)}. All cells m with |L_(m)|<K arediscarded. In the example, the only surviving cells are ab, b-d withL_(ab)={(1,2),(5,6)} and L_(b-d)={(2,4),(6,9)}

An exemplary embodiment of the present invention may also have aniteration phase. Let B be the collection of cells. If m=Extract(B), thenmεB and there does not exist m′εB such that m′

m holds: m₁

m₂ if one of the following holds: (1) m₁ has only solid characters andm₂ has at least one non-solid character (2) m₂ has the “-” character andm₁ does not, and, (3) m₁ and m₂ have d₁,d₂>0 dot characters respectivelyand d₁<d₂.

Further, m₁ is ˜-compatible with m₂ if the last solid character of m₁ isthe same as the first solid character of m₂.

Further if m₁ is ˜-compatible with m₂, then m=m₁˜m₂ is the concatenationof m₁ and m₂ with an overlap at the common end and start character and:L′ _(m)={((x, y), z)|((x,l), z)εL′ _(m) ₁ , ((l, y), z) εL′ _(m) ₂ }.  (42)

For example if m₁=ab and m₂=b.d then m₁ is ˜-compatible with m₂ andm₁˜m₂=ab.d. However, m₂ is not ˜-compatible with m₁.

An example, of this procedure is described by the pseudo-code shownbelow. NodeInconsistent(m) is a routine that checks if the new motif mis non-maximal w.r.t. earlier non-ancestral nodes by checking thelocation lists. Steps G: 18-19 detect the suffix motifs of alreadydetected maximal motifs. Result is the collection of all the maximalextensible patterns.

Main( )

Result←{ };

B←{m_(i)|m_(i)isacell};

For each m=Extract(B)

Iterate(m, B, Result);

Iterate(m, B, Result)

G:1 m′←m;

G:2 For each b=Extract(B) with

G:3 ((b˜-compatible m′) OR (m′˜-compatible b))

G:4 If (m′˜-compatible b)

G:5 m_(i)←m′˜b;

G:6 If NodeInconsistent(m_(i)) exit;

G:7 If (|L_(m′)|=|L_(b)|) B←B−{b};

G:8 If (|L_(m′)|≧K)

G:9 m′←m_(i);

G:10 Iterate(m′,B,Result);

G:11 If (b˜-compatible m′)

G:12 m_(t)←b˜m′;

G:13 If NodeInconsistent(m_(i)) exit;

G:14 If (|L_(m′|=|L) _(b)|) B←B−{b};

G:15 If (|L_(m′)|≧K)

G:16 m′←m_(t);

G:17 Iterate(m′, B, Result);

G:18 For each r ε Result with L_(r)=L_(m′)

G:19 If (m′ is not maximal w.r.t. r) return;

G:20 Result←Result∪{m′};

Correctness follows from the observation that the above exemplaryprocedure essentially constructs the inexact suffix tree of implicitly,in a different order. A tight time complexity is more difficult to comeby, however, if we consider M to be the number of extensible maximalmotifs and S to be the size of the output—i.e. the sum of the sizes ofthe motifs and the sizes of the corresponding location lists—then thetime taken by an exemplary embodiment of the present invention is O(SMlog M). In experiments by the inventors of the kind described below, at3 GHz clock, processing time ranged typically from few minutes to halfan hour.

A detailed description of an implementation of one exemplary embodimentin accordance with the present invention follows.

Since a pattern space can vary dramatically for different classes ofinputs, a number of parameters have been introduced to allow a usermaximally exploit his specific domain knowledge. One way of viewing thiscontrol is to prune the pattern space appropriately and variousparameters are specified to meet this objective. There are essentiallytwo classes of pruning parameters: (1) combinatorial and (2)statistical. To avoid clutter, we describe only a few of the pruningparameters here. Each parameter has a default value and it is notmandatory to specify them all.

Combinatorial Pruning

-   -   1. Pruning by Occurrences:        -   a. -k<Num>: Num is the quorum or the minimum number of times            a pattern must occur in the input.        -   b. -c: When this is specified the quorum k is in terms of            the number of sequences where the pattern occurs at least            once. For example, if this option is set and further -k10 is            specified, then a valid pattern must occur in at least 10            distinct sequences. However if this option is not set then a            valid pattern must have at least 10 occurrences, not            necessarily in distinct sequences.    -   2. Pruning by composition:        -   a. Using homology groups:            -   (1) -b<File>: File lists the symbol equivalences that                define the homology groups. The default file is an empty                file.            -   (2) -n<Num>: Num is the maximum number of bracketed                elements (equivalence classes) in a pattern. For                example, if “-n2” is specified, then [IL] . . . [LV],                L.[LV]−V are valid patterns but not [LV] [IL][LV] . . .                L.        -   b. -R: When this mode is specified, only rigid patterns are            discovered.        -   c. Extensibility: The following two parameters may be used            to prune the space of extensible patterns. FIG. 1 shows an            example of the size of the pattern space for different            parameter values.            -   (1) -D<Num>: Num is the maximum number of consecutive                don't care characters (‘.’) in the realization of an                extensible pattern. Note that a don't care character and                an extensible character are never consecutive in any                valid pattern. For example, if “-D3” is specified, then                L . . . V, LV, L.L.V are valid patterns but not L . .                . L. Further, an extensible pattern of the form L−V                implies that there are one to three don't care                characters in the occurrences of this pattern between                the bases L and V.            -   (2) -d<Num>: Num is the minimum number of non-extensible                characters (including the don't care character) between                two consecutive extensible characters (‘-’). For                example, if “-d4” is specified, then L . . . H-L . . .                H-L is a valid pattern but not L . . . H-L . . . H-L.

Statistical Pruning

1. -p<File>: File lists the symbol probabilities used for theprobabilistic analysis.

2. -z<Val>: Val is the minimum absolute value of Z-score of thepatterns.

Information Display

1. Displaying occurrence information: The different modes of displayingthe occurrence list of each valid pattern may be as follows. (1) Theoccurrence list is not displayed (option—L0). (2) Only the startposition of each occurrence is displayed (option—L1). (3) The start andend position of each occurrence is displayed as x₁-x₂ where x₁ is thestarting position and x₂ the end position(option—L4).

2. Displaying statistical information: The different statisticalinformation displayed for possible use are (1) the probability ofoccurrence of a pattern, (2) the observed number of occurrences, and (3)the Z-score. Table 1 shows an example.

Table 1. Numbers of patterns in the experiment in Table 8 withZ-Score≧≧100.0 at various values of parameters D and d with quorum k=53

D 2 3 4 5 d 3 121 196 370 1145 4 121 194 355 1008 5 114 182 326 891 8112 178 313 758 10 112 178 313 727

Pattern Probability Occ. Z-Score

[LIVP]-[LM]R.[GE][LIVP].GC 2.05647e−07 57 585.494

LR.[GE][LIVP].GC 2.53136e−07 63 582.758

L..[GE][LIVP].GC 4.77614e−06 70 148.626

R-[GE][LIVP].GC 6.33367e−06 66 121.48

L-[GE][LIVP].GC 1.43284e−05 83 101.21

G[LIVP][GE].GC 3.98344e−05 77 55.359

R-[LIVP].GC 4.68467e−05 65 42.6968

L-[LIVP].GC 0.00010598 112 48.3873

Table 2. A statistical summary of a small set of valid patterns on theCoagulation factors 5/8 type C domain, also used in Table 8.

FIG. 3 illustrates a flowchart 300 of an exemplary method in accordancewith the present invention. The flowchart 300 starts at step 302 andcontinues to step 304, where the method receives a sequence. Theflowchart continues to step 306, where the method assigns a significanceto an extensible motif within the sequence based upon a combination ofsyntactic and statistical analysis, an example of which is describedabove. The method continues to step 308 where the method identifies asignificant extensible motif by, for example, determining whether thesignificance assigned to an extensible motif exceeds a predeterminedthreshold. The method continues to step 310 where the system displays alist of the identified extensible motif from the sequence and continuesto step 312 where the method ends.

Experimental Results

The inventors tested an exemplary embodiment in accordance with thepresent invention on six protein families by seeking the surprisingmotifs in each. Each family was picked at random from the PROSITEdatabase.

-   -   High potential iron-sulfur proteins (HiPIP) (PROSITE I.D.        PS00596). This is a specific class of high-redox potential        4Fe-4S ferredoxins that function in anaerobic electron transport        and which occur in photosynthetic bacteria and in Paracoccus        denitrificans. Two of the cysteine residues of the motif shown        in Table 3 are involved in binding to the iron-sulfur cluster.        This is the top-ranking motif discovered by the exemplary        embodiment out of the possible 273 extensible motifs.    -   Streptomyces subtilisin-type inhibitors (PROSITE I.D PS00999).        Bacteria of the Streptomyces family produce a family of        proteinase inhibitors characterized by their strong activity        toward subtilisin. They are collectively known as SSI's:        Streptomyces Subtilisin Inhibitors. The exemplary embodiment        discovers this functionally significant motif as the top ranking        one out of 470 extensbile motifs (Table 4).    -   Nickel-dependent hydrogenases (PROSITE I.D PS00508). These are        enzymes that catalyze the reversible activation of hydrogen and        is farther involved in the binding of nickel. Again, this        functionally significant motif is detected in the top three by        the exemplary embodiment out of 4150 extensible motifs (Table        5).    -   G-protein coupled receptors family 3 (PROSITE I.D PS00980). The        exemplary embodiment finds that the most important structural        motif in this family is in the top thirty of the motifs out of        3508 extensible motifs (Table 6).    -   Chitin-binding type-1 domain (PROSITE I.D PS00026). The        exemplary embodiment finds that the most important structural        motif in this family is in the top two of the motifs out of 886        extensible motifs (Table 7).    -   Coagulation factors 5/8 type C domain (FA58C) (PROSITE I.D        PS01286). The exemplary embodiment finds that the most important        structural and functional motif in this family is in the top two        of the motifs out of 80290 extensible motifs (Table 8).

To summarize, the inventors discovered that in almost all cases, themotif documented as the most important (as functionally/structurallyrelevant motif) in PROSITE is in the top extensible motifs returned bythe exemplary embodiment as surprising. In the fourth set (Table 6) theinventors find the PROSITE motif at position 42, this experiment showsthat in some particular cases the patterns reported by the exemplaryembodiment can be grouped together, in fact the top scoring motifs arevery close to each other in location and composition. This reveals thata post-processing step that clusters together the top patterns mayimprove the results. In all cases, the difference in the z-score betweenthe top few and the rest is dramatic as can be seen in Tables 3 to 8.The differing values of the Z-scores of each family is attributed to thedifferent sizes of the families (the number of members and the length ofeach member).

The inventors also tested the sensitivity and selectivity of anexemplary embodiment of the present invention using the families asreported in PROSITE. The following six sets were selected by theinventors randomly in each family: 5 sequences in each of the families,high potential iron-sulfur proteins, streptomyces subtilisin-typeinhibitors, nickel-dependent hydrogenases, g-protein coupled receptorsfamily 3 and coagulation factors 5/8 type C domain, and 8 sequences fromthe family of chitin-binding type-1 domain.

First each family was contaminated with one of the sets that was drawnfrom a different family (for example the five sequences of G-protein wasmixed with the family of the hydrogenases). Next, the inventorscontaminated each family with two sets from a different family and thensubsequently three sets. In each of the experiments the inventorsdiscovered that the top ranked motifs were exactly as reported in Tables3 to 8.

Rank z-score Motif

1 1497,62 C-(6,7,8,9)[LIVM] . . . G[YW]C . . . [FYW]

2 978,872 P-(3,4,6,8,9)[LIVM] . . . G[YW]C . . . [FYW]

3 590,866 C-(6,7,8,9)[LIVM] . . . G[YW]C-(1,3,4,5,6,7)A

4 564,821 C-(6,7,8,9)[LIVM] . . . G[YW]C-(1,3,4,5,6,7)[ATD]

5 537,73 [LIVM]-(1,2,3,4,5,7,8,9)G[YW]C . . . [FYW]

6 385,2 [LIVM]-(1,2,3,4,5,7,8,9)G[FYW]C . . . [FYW]

7 161,173 [LIVM] . . . G[FYW]C-(2,4)[FYW]

8 156,184 [LIVM]-(1,2,3,4,5,6,7,8,9)G[YW]C

9 138,881 [LIVM]-(1,3,4,5,6)[LIVM] . . . G[FYW]C-(1,3,4,5,6,7)A

Table 3. The functionally relevant motif is shown in bold for highpotential iron-sulfur proteins (HiPIP) (id PS00596). Here 22 sequencesof about 2500 bases were analyzed at k=22, D=9, d=4.

Rank z-score Motif

1 7,60E+07 RA.T[LV].C.P-(2,3)G.HP . . . AC[ATD].L . . . [ASG]

2 21416,8 A . . . [LV].C.P-(2,3)G.HP-(1,2,4)[ASG].[ATD]

3 8105,33 A-(1,4)T . . . P-(2,3)G.HP . . . [ATD]-(3)L . . . [ASG]

4 5841,85 [ATD].T . . . P-(1,2,3)G.HP-(1,2,4)A.[ATD]

5 4707,62 P.[ASG]-(2,3,4)P . . . AC[ATD].L . . . [ASG]

6 4409,21 A . . . [LV] . . . P-(2,3)G.HP-(1,2,4)A.[ATD]

7 3086,17 P-(1,2,3)[ASG] . . . P-(4)AC[ATD].L . . . [ASG]

8 3068,18 R . . . [ATD] . . . P-(2,3)G.HP-(1,2,4)[ASG].[ATD]

9 2615,98 [ASG][ATD]-(1,3,4)P . . . AC[ATD].L . . . [ASG]

10 2569,66 [ASG]-(1,2,3,4)P . . . AC[ATD].L . . . [ASG]

11 2145,6 G-(2,3)P . . . AC[ATD].L . . . [ASG]

Table 4. The functionally relevant motif is shown in bold forStreptomyces subtilisin-type inhibitors signature (id PS00999). Here 20sequences of about 2500 bases were analyed at k=20, D=4, d=4.

Rank z-score Motif

1 295840 [LIM]-(1,2,3,4)[STA][FY]DPC[LIM][ASG]C[ASG].H

2 2,86E+05 [LIM]-(1,2,3,4)[ASG][FY]DPC[LIM][ASG]C[ASG].H

3 155736 R-(1,4)[FY]DPC[LIM][ASG]C[ASG].H

4 78829 [LIM]-(1,2,3,4)[STA].DPC[LIM][ASG]C[ASG].H

5 76101,9 [LIM]-(1,2,3,4)[ASG].DPC[LIM][ASG]C[ASG].H

6 34205,6 [STA]-(1,4)DPC[LIM][ASG]C[ASG].H

7 30325,1 [LIM]-(1,2,3,4)[STA][FY]D.C[LIM][ASG]C . . . H

8 29276 [LIM]-(1,2,3,4)[ASG][FY]D.C[LIM][ASG]C . . . H

9 20527,3 [ASG]-(1,4)DPC[LIM][ASG]C[ASG].H

10 17503,4 [LIM]-(1,2,3,4)[ASG] . . . PC[LIM][ASG]C[ASG].H

Table 5. The functionally relevant motifs are shown in bold forNickel-dependent hydrogenases (PROSITE I.D. PS00508). Here 22 sequencesof about 23000 bases were analyzed at k=22, D=4, d=3.

Rank z-score Motif

1 2,84E+09 Y . . . L . . . C . . . [FYW]A . . . [STAH]R . . . P . . .FNE[STAH]K.I.F[STAH]M

2 8,28E+07 V-(1,3,4)G . . . S . . . [STAH] . . . N . . . L . . .Q-(4)[STAH] . . . L.[DN] . . . [FYW] . . . F . . . P . . . Q . . . A . .. I

3 5,55E+07 L-(2,3)F . . . Q . . . [STAH][STAH] . . . L.[DN] . . . [FYW].. . F.R . . . P.D . . . Q . . . A . . . I

4 4,27E+07 L-(2,3)F . . . Q.[STAH] . . . [STAH][STAH] . . . S . . .[FYW] . . . F.R . . . P.D . . . Q . . . A . . . I

5 4,23E+07 L . . . I . . . [STAH] . . . [STAH] . . . LS[DN] . . . [FYW]. . . F.R . . . P.D . . . Q . . . A . . . I

6 3,99E+07 LF-(3)Q . . . [STAH][STAH] . . . . S[DN] . . . [FYW] . . .F.R . . . P.D . . . Q . . . A . . . I

7 3,38E+07 LF-(3)Q . . . [STAH][STAH] . . . L.[DN] . . . [FYW] . . . F.R. . . P.D . . . Q . . . A . . . I

8 3,38E+07 LF . . . Q . . . [STAH]-(4)L.[DN] . . . [FYW] . . . F.R . . .P.D . . . Q[STAH].A . . . I

9 3,29E+07 I-(1)Q.[STAH] . . . [STAH] . . . LS[DN] . . . [FYW] . . . F.R. . . P.D . . . Q . . . A . . . I

10 3,29E+07 I.Q-(4)[STAH] . . . LS[DN] . . . [FYW] . . . F.R . . . P.D .. . Q[STAH].A . . . I

11 3,29E+07 I.Q.[STAH] . . . [STAH]-(4)LS[DN] . . . [FYW] . . . F.R . .. P.D . . . Q . . . A . . . I

12 3,10E+07 L . . . Q-(1,4)[STAH] . . . [STAH] . . . LS[DN] . . . [FYW]. . . F.R . . . P.D . . . Q . . . A . . . I

13 2,77E+07 L[FYW]-(3)Q.[STAH] . . . [STAH] . . . LS . . . [FYW] . . .F.R . . . P.D . . . Q . . . A . . . I

14 2,58E+07 L-(4)Q.[STAH] . . . [STAH] . . . LS[DN] . . . [FYW] . . .F.R . . . P.D . . . Q . . . A . . . I

15 2,30E+07 S.[STAH]S-(2,4)LS[DN] . . . [FYW] . . . F.R . . . P.D . . .Q[STAH].A . . . I

16 2,15E+07 L-(1,3,4)C . . . [FYW]A . . . [STAH]R . . . P . . .F.E.K.I.F.M

17 1,40E+07 F-(1)I.Q . . . [STAH][STAH]-(4)L[STAH] . . . [FYW] . . . F.R. . . P.D . . . Q . . . A . . . I

18 1,37E+07 L-(2,4)I . . . [STAH].[STAH].[STAH]-(3)LS . . . [FYW] . . .F.R . . . P.D . . . Q . . . A . . . I

19 1,02E+07 L . . . I-(1)Q . . . [STAH][STAH] . . . S . . . [FYW] . . .F.R . . . P.D . . . Q . . . A . . . I

20 8,65E+06 I-(1)Q . . . [STAH][STAH] . . . L.[DN] . . . [FYW] . . . F.R. . . P.D . . . Q . . . A . . . I

21 8,19E+06 S[STAH]-(1,2,3,4)LS[DN] . . . [FYW] . . . F.R . . . P.D . .. Q[STAH].A . . . I

22 7,98E+06 Q-(3)[STAH][STAH] . . . LS[DN] . . . [FYW] . . . F.R . . .P.D . . . Q . . . A . . . I

23 6,82E+06 F-(3)Q . . . [STAH][STAH] . . . L[STAH] . . . [FYW] . . .F.R . . . P.D . . . Q . . . A . . . I

24 5,66E+06 A[STAH][STAH]-(2,3)LS[DN] . . . [FYW] . . . F.R . . . P.D .. . Q . . . A . . . I

25 5,57E+06 F.I-(3)[STAH] . . . [STAH] . . . L[STAH] . . . [FYW] . . .F.R . . . P.D . . . Q . . . A . . . I

26 5,18E+06 L.L-(4)Q . . . [STAH] . . . L-(1)[DN] . . . [FYW] . . . F.R. . . P.D . . . Q . . . A . . . I

27 3,61E+06 L.L-(2)I . . . [STAH] . . . [STAH] . . . . [STAH]. . . .[FYW]. . F.R . . . P.D . . . Q . . . A . . . I

28 3,48E+06 [STAH]. [STAH]-(1,2,3)LS[DN] . . . [FYW] . . . F.R . . . P.D. . . Q . . . A . . . I

29 3,17E+06 [STAH] . . . [STAH] . . . LS[DN] . . . [FYW] . . . F.R . . .P.D . . . Q . . . A . . . I

30 2,47E+06 L . . . Q-(4)[STAH][STAH] . . . S . . . [FYW] . . . F.R . .. P.D . . . Q . . . A . . . I

31 2,43E+06 V-(1,3)N.L . . . I-(3)[STAH] . . . [STAH] . . . [STAH] . . .[FYW] . . . F . . . P.D . . . Q . . . A . . . I

32 2,22E+06 [STAH][STAH][STAH]-(1,2,3)LS . . . [FYW] . . . F.R . . . P.D. . . Q . . . A . . . I

33 2,06E+06 [STAH].[STAH][STAH] . . . LS . . . [FYW] . . . F.R . . . P.D. . . Q . . . A . . . I

34 2,03E+06 Y . . . L . . . C . . . A . . . R . . . P . . .F.E.K.I-(1,4)[FYW][STAH]

35 1,99E+06 I.Q . . . [STAH]-(1)[STAH] . . . L.[DN] . . . [FYW] . . . F. . . P.D . . . Q . . . A . . . I

36 1,99E+06 I.Q-(1)[STAH] . . . [STAH] . . . L.[DN] . . . [FYW] . . . F. . . P.D . . . Q . . . A . . . I

38 1,97E+06 F.I . . . [STAH]-(3)[STAH] . . . L.[DN] . . . [FYW] . . . F. . . P.D . . . Q . . . A . . . I

40 1,97E+06 F.I-(3)[STAH] . . . [STAH] . . . L.[DN] . . . [FYW] . . . F. . . P.D . . . Q . . . A . . . I

41 1,91E+06 [STAH] . . . [STAH] . . . K-(1,4)P . . .FNE[STAH]K.I.F[STAH]M

42 1,72E+06 CC[FYW].C . . . C . . . [FYW]-(2,4)[DN] . . . [STAH]C . . .C

43 1,57E+06 [STAH]-(1,3,4)[FYW]A . . . [STAH]R . . . P . . . F.E.K.I.F.M

44 1,49E+06 A-(1,3)[STAH] . . . L[STAH][DN] . . . [FYW] . . . F.R . . .P.D . . . Q . . . A . . . I

45 1,36E+06 Q . . . [STAH].[STAH]-(3)L[STAH] . . . [FYW] . . . F.R . . .P.D . . . Q . . . A . . . I

46 1,32E+06 I-(3)[STAH] . . . [STAH][STAH] . . . S . . . [FYW] . . . F.R. . . P.D . . . Q . . . A . . . I

47 1,31E+06 [STAH][STAH]-(1,2,3,4)L.[DN] . . . [FYW] . . . F.R . . . P.D. . . Q . . . A . . . I

48 1,24E+06 [STAH] . . . [STAH][STAH]-(1,3)LS . . . [FYW] . . . F.R . .. P.D . . . Q . . . A . . . I

49 1,19E+06 [FYW]-(1,3,4)[STAH] . . . P . . . FNE[STAH]K.I.F[STAH]M

50 1,12E+06 I . . . [STAH]-(3)[STAH] . . . L[STAH] . . . [FYW] . . . F.R. . . P.D . . . Q . . . A . . . I

Table 6. The functionally relevant motif is shown in bold for G-proteincoupled receptors family 3 (PROSITE I.D. PS00980). This run involved 25sequences of about 25000 bases each at k=25, D=4, d=8.

Rank z-score Motif

1 5,42E+06 C-(4,5)CCS . . . G[FYW]CG . . . [FYW]C

2 1,73E+06 C-(4,5)CCS . . . G[FYW]CG . . . C

3 1,70E+06 C-(4,5)CCS . . . G.CG . . . [FYW]C

4 1,56E+06 CCS . . . G[FYW]CG . . . [FYW]C

5 544162 C-(4,5)CCS . . . G.CG . . . C

6 4,95E+05 CCS . . . G[FYW]CG . . . C

7 488261 CCS . . . G.CG . . . [FYW]C

8 155706 CCS . . . G.CG . . . C

9 104666 C-(4,5)C.S . . . [GASL][FYW]CG . . . C

10 84133,4 C . . . C-(3,4)[GASL][FYW]CG . . . [FYW]C

11 56078 C . . . C-(3,4)G.CG . . . [FYW]C

Table 7. The functionally relevant motif is shown in bold for Chitinrecognition (PROSITE I.D. PS00026). Here 53 sequences of about 13823bases were analyzed at k=53, D=5, d=10.

Rank z-score Motif

1 969,563 P-(4,5,8,9,10)[LM]R.[GE][LIVP].GC

2 694,1 P-(4,5,8,9,10)[LM]R.[GE][LIVP].[GE]C

3 370,594 [LIVP]-(1,3,4,5,6,7,8,9,10)[LM]R.[GE] . . . [GE]C

4 361,052 P-(4,5,8,9,10)[LM]R.[GE] . . . [GE]C

5 261,519 [LIVP]-(1,3,4,5,6,7,8,9,10)[LM]R.[GE][LIVP] . . . C

6 261,519 [LIVP]-(1,3,4,5,6,7,8,9,10)[LM]R . . . [LIVP].[GE]C

7 254,971 P-(4,5,8,9,10)[LM]R.[GE][LIVP] . . . C

8 254,971 P-(4,5,8,9,10)[LM]R . . . [LIVP].[GE]C

9 249,763 [LIVP] . . . [LIVP]-(1,2,4,5,6,7,8,9,10)R.[GE] . . . GC

Table 8. The functionally relevant motif is shown in bold forCoagulation factors 5/8 type C domain (PROSITE I.D. PS01286). Here 40sequences of about 80290 bases were analyzed. Notice that in this case,the motifs have a fairly large gap size of 10 bases at k=40, D=10, d=10.

The extensibility of a motif not only leads to a succinct descriptionbut also helps capture function and/or structure in a single pattern,which would be not possible through a rigid description. At the sametime, with extensible motifs the number of candidates to be consideredincreases dramatically.

An exemplary embodiment of the present invention characterizes a patternrigidly, and conjugates structure and set of occurrences. This resultsin a definition of motif that lends itself to a natural notion ofmaximality, thereby embodying statistics and structure in one measure ofsurprise. This is unlike all previous approaches that consider structureand statistics as separate features of a pattern.

An exemplary embodiment of the present invention provides a powerfulsyntactic mechanism for eliminating unimportant motifs before theirscore is computed. As explained above, for the class of over-representedmotifs, the non-maximal motifs are not more surprising than the maximalmotifs.

While the invention has been described in terms of several exemplaryembodiments, those skilled in the art will recognize that the inventioncan be practiced with modification.

Further, it is noted that, Applicant's intent is to encompassequivalents of all claim elements, even if amended later duringprosecution.

What is claimed is:
 1. A method of extracting an extensible motif from asequence, said method comprising: assigning on a computer a significanceto an extensible motif within said sequence based upon a syntactic andstatistical analysis; identifying an extensible motif having asignificance that exceeds a predetermined threshold; and iterating saidassigning said significance and said identifying said extensible motif,wherein said assigning said significance comprises assigning saidsignificance based on a combination of saturation conditions at eachiteration and monotonicity of probabilistic scores derived from thesaturation conditions of said extensible motif.
 2. The method of claim1, wherein said extensible motif comprises at least one annotated dotcharacter.
 3. The method of claim 1, further comprising displaying alist of identified extensible motifs.
 4. The method of claim 3, furthercomprising displaying statistical information for each of said displayedextensible motifs.
 5. A system for extracting an extensible motif from asequence, said system comprising: means for assigning a significance toan extensible motif within said sequence based upon a syntactic andstatistical analysis; means for identifying an extensible motif having asignificance that exceeds a predetermined threshold; and means foriterating said assigning said significance and said identifying saidextensible motif, wherein said means for assigning said significancecomprises means for assigning said significance based on a combinationof saturation conditions at each iteration and monotonicity ofprobabilistic scores derived from the saturation conditions of saidextensible motif.
 6. The system of claim 5, wherein said extensiblemotif comprises at least one annotated dot character.
 7. The system ofclaim 5, further comprising means for displaying a list of saididentified extensible motifs.
 8. A non-transitory computer-readablestorage medium encoded with a computer program, said program comprising:instructions for assigning a significance to an extensible motif withinsaid sequence based upon a syntactic and statistical analysis;instructions for identifying an extensible motif having a significancethat exceeds a predetermined threshold; and instructions for iteratingsaid assigning said significance and said identifying said extensiblemotif, wherein said assigning said significance comprises assigning saidsignificance based on a combination of saturation conditions at eachiteration and monotonicity of probabilistic scores derived from thesaturation conditions of said extensible motif.